117
Netherlands Journal of Sea Research
20 (2/3):
117-131 (1986)
TIDAL ASYMMETRY AND ESTUARINE MORPHOLOGY
J.DRONKERS
Divisionof TidalWaters,Rijkswaterstaat,v. Alkemadelaan400, 2597AT TheHague,TheNetherlands
ABSTRACT
Estuarine morphology is to a large extent determined by the residual sediment transport pattern.
However, the inverse statement is also true.
Residual sediment transport depends on differences in magnitude and duration between ebb
and flood tidal currents. Such differences ("tidal
asymmetry") are produced by the distortion of
the tidal wave propagating on the coastal shelf
and entering bays and estuaries. In this study the
relationship between tidal asymmetry and estuarine morphology is investigated. Based on theoretical considerations some general principles
are
derived
and
compared
with field
observations.
1. INTRODUCTION
The evolution of an estuary depends essentially on
two processes:
- the long-term averaged sediment supply from inland or coastal origin, and the direction and
magnitude of the long-term averaged sediment
transport,
- abrupt changes in the estuarine morphology
caused by storm surges or by engineering works.
The present study is concerned with the first process. The second process will be invoked when
some practical examples are discussed.
It is often conjectured that estuaries tend to fill in
with sediment and will, therefore, ultimately disappear. In the final stage of evolution, river water is
discharged directly on the coastal shelf (SCHUBEL
&
MEADE, 1977). However, temporarily certain
estuaries may behave as erosion basins, as a consequence of externally induced changes in sediment
supply or flow regime. Whether stable estuaries exist
(sediment inflow on the average balanced by sediment outflow) is not known, and it also seems an
academic question: there always exist external con-
ditions (e.g. the mean sea level) subject to variations
(POSTMA,1980).
The sediment supply and sediment transport pattern in an estuary depend on several factors, some
of which will be discussed in more detail:
-(a) The river inflow, with related to it (a1) an input
of sediment and (a2) a cross-sectional flow structure
affected by density differences. Density differences
tend to increase flood currents in the deepest parts of
the channel, in particular in the lower half of the vertical, and tend to decrease flood currents near the
surface and in the shallow parts of the cross-section.
The inverse holds for the ebb currents.
This flow structure has an important impact on the
sediment transport. It has been discussed already by
many authors, for example POSTMA
(1967), FESTA&
HANsEN(1978), ODD & OWEN(1972), ALLENet al.
(1977), ARIATHURAI
et al. (1977). The present study
will not enter further into details.
-(b) The sediment characteristics. In general a
wide spectrum of sediment types is present in
suspension at the same time. In this study only a
crude distinction between "fine" and "coarse" sediment will be made, based mainly on fall velocity w
(for "coarse" sediment w ~ 10-1 m.s -1, for fine
sediment W:$ 10 - 2 m.s - 1, see also section 2). For
both sediment types two transport modes exist:
along the bottom (coarse grains moving by traction
in the bedload, fine sediment moving as fluid mud),
or as suspended load. As will be argued in section 2,
tidal asymmetry affects suspension transport in particular. The influence of tidal asymmetry on the
residual fluxes of coarse and fine sediment is dif.
ferent, owing to different transport properties: the
suspension load of coarse sediment is strongly
limited by the current speed and it adapts to changes
in the current speed rapidly in comparison with the
tidal time scale. For fine sediment, saturation of the
suspended load seldom occurs. Most fine sediment
settles only at very Iow current speed and the settling
time delay is important.
118
J. DRONKERS
-(c) Wind waves, swell.The purelywave-induced
sediment transport is relatively less important in
estuaries than on the coastal shelf. However, the influence of wind waves, in tidal flat areas at high
water, on the resuspension of sediment is a significant phenomenon (ANDERSON,
1972; McDoWELL&
O'CONNOR,1977). In combination with subsequent
tidal transport, it can cause a substantial seaward
sediment flux.
-(d) The current velocity distribution and in particular its variation during a tidal cycle. Twoelements are
mainly responsible for residual sediment fluxes (in
addition to density influences):
-(d1)The tidal variation at the sea entrance, which
bears the characteristics of tidal wave propagation on
the coastal shelf (amplification, distortion),
-(d2) The tidal propagation inside the tidal basin,
generally consisting of a complex geometrical
system formed by meandering and braiding channels
and tidal flats.
The present study is mainly devoted to aspect (d): the
analysis of tidal wave deformation in shallow systems
with a regular or a complex geometry, and its impact
on the residual sediment flux. Globally speaking the
following features of tidal wave deformation are relevant for residual sediment transport (a more detailed
discussion is presented in section 2):
- a difference between the maximum tidal currents
during ebb and flood, which in particular affects the
residual flux of the coarse suspended fraction,
- a difference between the slack water periods preceding ebb and flood, which particularly influences
the residual flux of the fine suspended fraction.
In sections 3 and 4 it will be shown that a tidal wave
changes its original sinusoIdal shape and becomes
asymmetric when it enters a tidal basin:
- if the mean water depth is so small that the tidal
variation cannot be neglected; the geometry changes
significantly with tidal level, or
- if the velocity field changes significantly over a
distance comparable with the tidal excursion (for
example, the channel geometry varies over such
distances).
In addition, the sinuso'idal shape of a tidal wave is
altered by the quadratic dependence of bottom friction on the current speed. However,this quadratic character conserves the ebb-flood symmetry, and no residual sediment flux results. Also, the main characteristics of tidal distortion will be demonstrated on a
simplified model with bottom friction depending linearly on the current velocity. In the appendix, first
order approximations of the tidal wave deformation
due to different non linear terms are presented for a
uniform rectangular basin with and without wave
reflection.
Section 2 deals with the influence of tidal asymmetry on the residual sediment flux. In sections 3 and 4,
the propagation and distortion of a tidal wave in different types of basins is discussed: tidal rivers with no
substantial wave reflection (section 3) and almost cooscillating short tidal basins with a complex geometry (section 4). In section 5 field observations of the
residual sediment flux in two tidal basins in the Netherlands are discussed as an example. In section 6
the principal results are briefly summarized.
Many of the ideas today forming the foundation of
the geomorphological science of estuarine and
coastal environments have been developed by Henk
Postma. His sciehtific work has been an important
source of inspiration for this study.
Acknowledgements.-Numerical model simulations
were performed by M. Geurtz in order to check the
general validity of several statements on the relationship between tidal asymmetry and estuarine morphology. Improvements of the draft have been suggested by J.R. van der Berg, L. Kohsiek and C.
Louisse. The figures were prepared by Th. Vogel and
the typewriting by Mrs E. Goldbach.
2. SEDIMENT TRANSPORT DUE TO
TIDAL ASYMMETRY
This section contains some general considerations
on the residual flux of sediment in tidal environments.
Successively, bed load transport and suspension
transport will be discussed for both coarse and fine
sediment.
Bed load transport of coarse sediment consists
of grains with a diameter in the order of 200 J.lmor
larger, rolling or jumping over the bed (YALlN,1972;
McDoWELL& O'CONNOR,1977;HEATHERSHAW,
1981).
The velocity of such grains is much smaller than the
current speed. As a consequence the local dominance of ebb or flood currents mainly determines the residual bed load flux. A general feature of sandy bays
and estuaries is the presence of a pattern of alternating ebb and flood channels (VANVEEN,1950; ROBINSON, 1960). As stated by LAMBIASE(1980), slowmoving grains transported by traction are trapped in
such channel systems. (On the contrary, suspended
grains generally move fast enough to bypass channel
sections where locally the residual current is opposed). The conclusion is that in most sandy estuaries
an ebb-flood asymmetry of the tidal wave does not
greatly affect the residual flux of bed load sediment.
Transport of fine sediment along the bottom is ge-
119
TIDAL ASYMMETRY AND ESTUARINE MORPHOLOGY
nerally described as "fluid mud". This phenomenon
has been observed in several estuaries (e.g.the Thames and the Gironde) and is associated with the occurrence of a turbidity maximum (INGLlS& ALLEN,
1957;ALLENet al., 1977).In these estuaries a distinct
ebb-flood asymmetry in the near-bottom layer is caused by estuarine gravitational circulation. Again, an
additional ebb-flood asymmetry due to tidal wave distortion is not the major factor for residual transport,
although it may have a quantitatively significant
influence.
Suspension transport of coarse sediment possesses the following characteristics: the maximum sediment load in suspension at a given current speed is
limited and depends on particle size and density
(BRUUN,1978;DYER,1980).The saturation load is rapidly reached (as compared to the tidal time scale).
Commonly found sediment particles with these
transport characteristics in the estuarine environment are fine sands (size on the order of 100j.lm)and
large biologically bound aggregates (size on the
order of a few hundred j.lm) (BIGGS, 1978; POSTMA,
1980; EISMAet al., 1980). The saturation load increases very strongly with increasing current velocity. Therefore, a tidal asymmetry in the current velocity variation may affect the residual transport of coarse
sediment. The most significant asymmetry in this
respect is a difference between the maximum current
velocities occurring during flood and ebb. For example, if flood velocities exceed ebb velocities in compensation of a longer ebb duration, then a residual
landward transport of coarse sediment will exist.
In many estuaries, the suspended load is mainly
composed of fine sediment: particles with a size between 1 and 10 j.lm (quartz, feldspar, clay minerals)
and aggregates with a size up to 100 j.lm (MEADE,
1972; EISMAet al., 1980). The transport of this
material is more complicated than the transport of
coarse sediment. This is due in particular to the
cohesive properties (consolidation and flocculation)
and related time lag effects. Even at rather small current velocities (a few dm.s -1), an important load of
fine sediment can be maintained in suspension. Actually, saturation seldom occurs. The concentration
of fine suspended sediment is limited by the availability of erodible bottom material, i.e. by consolidation or
by the presence of an overlying coarse sediment
layer. Weakly consolidated fine sediment is
resuspended when the current velocity reaches a
critical value Ue (see, for example, CREUTZBERG
&
POSTMA, 1979; MEHTA & PARTHENIADES,
1982;
DRONKERS,1985). When the current velocity increases further, more consolidated fine sediment and
deeper layers may be brought in suspension.
Part of the fine sediment load settles in the period
around slack water, in particular the aggregates with
a size on the order of 100 j.lm and larger (POSTMA,
1960; CHASE,1980).Adhesion on the bottom may occur when the current speed has dropped below a
threshold value Ud,Ud< ue (EINSTEIN
& KRONE,1962).
It follows that the fine sediment load responds more
strongly to tidal variations in the period around slack
water than around maximum current speed. The
slack water periods will be designated as SBE and
SBF: Slack tide Before Ebb (seaward flow) and Slack
tide Before Flood (landward flow), respectively.
For the residual transport of fine suspended sediment, an approximate analytical expression has
been derived by DRONKERS
(1985),based on a qualitative description by POSTMA
(1961).In this derivation
the excursion of sediment particles through the estuary is followed during a tidal cycle. Relevant quantities are the water depth and the current velocity variation, which are compared at t+ (=SBE) at a location x+ (=the landward limit of the tidal excursion),
and at t- (=SBF) at a location x- (=the seaward limit). The resulting expression for the net transported
sediment mass through a cross section x ==
'12(x + + x-) during a tidal cycle reads:
M=j.I+).+
(1)
-j.l-).-
where j.I+ (respectively j.I-) is the amountof sediment settled on the bottom at x + (respectively x-) in
the period of SBE (respectively SBF), and). +
(respectively). -) is the distance travelled by fluid
parcels in the SBE (respectively SBF) time interval
~t + (respectively ~t-) during which the fine fraction
remainssettled.The quantitiesj.I::!:and),:!:
can be
evaluated from the approximate expressions:
j.I:!:(x)==U):!:(x).max. susp. conc. (x)
w:!:(x)==A s (x:!:, t:!:)[1-exp(-w
M :!:(x:!:)
d
)1 (2)
h(x:!:, t:!:)
w = fall velocity,As = stream cross-section,
interval for which I u I < ud'
).:!: (x)
==
1!2ue.~t:!: (x:!:)
~td = time
(3)
The physics behind these equations is obvious: the
amount of sediment j.I+, which is settled per unit
length in the period around SBE, will not follow the
tidal motion before the ebb current reaches the critical speed for erosion ue' In this lapse of time the
J. DRONKERS
120
settled sediment is displaced with respect to the
suspended sediment in landward direction over a distance which on the average equals A+ . Around SBF
a similar relative displacement will occur of a sediment mass J.I- in seaward direction over an average
distance A-.
The equations (2) and (3) show that a landward residual flux of fine sediment is favoured if:
- the channel depth decreases in landward direction, h(x+, t+)<h(x-, t-),
- the velocity variation is slower around SBE than
around SBF,
3. DISTORTION OF A NON-REFLECTED
TIDAL WAVE
The cross-sectionally integrated tidal equations, valid
for wide shallow basins (b:ph), read:
~dt +! b dx
~ (Asu)=O
(4)
<!.':'+u<!.':'+g~+F~
=0
dt
dx
dx
h+~
(5)
Here the total cross-section A (width at the water
surface=b) has been divided into two parts. The entire transport takes place in one part (cross
This last condition can be realized either by a distorti- section=As' average depth=h). The other part
on of the tidal wave (see the sections 3 and 4) or by (which is shallow, the depth averaged velocity being
a landward decrease of the current velocity. The lat- less than, about, 0.3 u) is considered to act as a storater aspect has been demonstrated in particular by ge area only. The friction term is not quadratic but liVAN STRAATEN & KUENEN (1959).
near with the current velocity. As explained, essentially this simplification does not affect the ebb-flood
Finally the influence of wind wavesshould be men- asymmetry of the tidal wave.Typicalvalues of the frictioned. In addition to tidal currents, short waves also tion coefficient F range between 10-3 and 5.10-3
contribute to bringing and keeping sediment in m.s-1.
suspension. Wave energy dissipation is mainly
In this section long inshore tidal basins or tidal rirestricted to shallow regions, with water depth in the vers are considered in which the tidal wave can proorder of a few meters or less. In estuaries with land- pagate without substantial reflection. In practice, this
ward decreasing depth, the amount of sediment de- implies a geometry without important cross-sectional
posited in the period around SBE is more strongly variations. If a sufficiently regular geometry is assudecreased by wave action than the amount deposited med, the equations (1) and (2) can be simplified to
around SBF. In such estuaries wind wavescounteract the equations (A1)and (A2) of the appendix. In these
a landward residual transport of fine sediment by tidal equations three non-linear terms are responsible for
the distortion of the tidal wave.The influence of these
currents, or enhance a seaward residual transport.
terms is demonstrated in the appendix and the Figs
The seaward flux due to wind waves can be parti- 10a to c. A qualitative discussion is presented below.
cularly important in estuaries with high tidal flats (siIn the continuity equation (A1)the term ~du/dx retuated above mean sea level). In a meandering chan- presents the tidal variation of water depth. On the
nel systemsuch high tidal flats are built up by a near- average the water depth is larger during flood than
bottom flow component which is directed up slope at during ebb. Therefore, a residual discharge in the wathe headlands during both ebb and flood (HEATHER- ve propagation direction will exist unless the flood
SHAW& HAMMOND,
1980). The fine sediment eroded currents are smaller than the ebb currents. This phefrom the tidal flats by waves in the period around high nomenon is usually described as "Stokes drift".
water remains in suspension during most of the ebb
In the momentum equation (A2) the term udu/dx
period and is, therefore, transported a long distance constitutes a contribution to the acceleration duldt,
in seaward direction. Coarse sediment is also which tends to increase the magnitude of the accelebrought in suspension by wind waves. It will be depo- ration during flood and to decrease the magnitude
sited sooner than the fine sediment and will thus be during ebb. The result is contrary to the Stokes drift:
transported less far by the ebb current. However,this a relatively shorter flood period with relatively higher
coarse sediment is not so easily resuspended by the velocities, as compared to the ebb period. However,
flood current: in tidal flat estuaries wind waves also Stokes drift is the dominating effect, see Fig. 10a.
cause a net seaward transport of coarse sediment. In
A very important aspect of the non-linear terms
other words: tidal flat formation by tidal currents, udMdx and udu/dx is to introduce total time derivaticounteracted by wind wavesmay finally lead to an ex- ves d/dt instead of partial derivates d/dt,
port of sediment.
d/dt= d/dt+ud/dx. For an observer moving with the
I du/dt I x + ,t + < I du/dt I x- . t -
TIDAL ASYMMETRY AND ESTUARINE MORPHOLOGY
current velocity u the tidal equations remain linear
(except for the term ~du/dx).As a consequence, the
wave crest and wave trough propagate at different
velocities, c + umaxand c - umaxrespectively. Here
c=
and umax== ac/h.Theterm~du/dxyieldsa larger water depth for wave propagation in the crest
than in the trough V'g(h+ a) and V'g(h- a) respectively. Combining both effects the wave crest moves
~
(1 + ~~) and the
through with velocity ~ (1 - ~~ ). As a result, the
rising part of the wave surface becomes steadily
steeper, and the falling part steadily flatter, see Fig.
10a. This result was established long ago by AIRY
(1842). Ultimately this deformation leads to a tidal
bore and wave breaking.
A tidal bore develops in an estuary if the tidal range
b.~is sufficiently large and if the bottom slope Ib in
the zone situated around mean coastal sea level is
sufficiently gentle (COMOY,1881; LYNCH,1982)
~~:z: ~lb'T..J9(h+ <~»
See Fig. 1a. Well-known examples are the rivers
TSientang,Severn and Hooghly.
In most tidal rivers this final stage of distortion is
not reached. In general only a slightly steeper rise
than decrease of water levels is observed. Accordingly, the slack water period preceding flood is longer
than the slack water period preceding ebb. Examples
are shown in Fig. 2. As discussed in section 2, import
of fine sediment is favoured by this tidal asymmetry
over export.
From the analysis of IWAGAKI
& SAKAI(1974)it follows that the above discussion of wave deformation
(J
CURRENT
TIDE
LEVEL
VELOCITY
[rn/s ]
.,O
FLOOD
~
approximately with velocity
121
Iml
j
/
.0.5
f\
~
0
t
EBB
-0.5
T
'2
.1.0
.1
.0.5
0
0
-1
-0.5
-2
-1.0/-1.5
.2
".0
'1
.0.5
0
FLOOD
0
-1
-2
-3
-1.5/.2.0
.3
'15
-2
01.0
.1
.0.5
0
0
-1
-0.5
-2
-1.0
-3
'1.0
FLOOD
.O.~
0
-0.5
0
4
B
12
EBB
-1.0
16 [hours J
fig. 2. field observations of tidal wave propagation in rivers.
Ib
"EAN
SEA LEvEL
TIDAL
c)
.- - - - -~
~
.
RISING
FLAT
TIDE
- -
H-'-1_LLlLLfIif'i
/1//11/1111/1/1/1'
b)
FALLING
m
TIDE
Fig. 1.Tidal propagation on a gently sloping tidal flat. a. formation of a bore during rising tide. b. Development of a
strong surface inclination during ebb.
by non-linear terms remains qualitatively valid if, instead of a constant depth, a slowly decreasing depth
in the wave propagation direction is considered,
which is in general more realistic for tidal rivers.
A particular aspect of the previous discussion is
the tidal wave distortion in shallow coastal seas.
Along the western coast bf the Netherlands a semidiurnal tidal wave propagates from south to north, belonging to the amphidromic system in the Southern
Bight of the North Sea. Along the northern coast of
the Netherlands a semidiurnal tidal wave propagates
from west to east, belonging the amphidromic system
of the central North Sea. Both tidal waves are distorted when running along the Dutch coast, and on the
way exhibit an increasingly stronger rise and slower
122
J. DRONKERS
fall, see Fig. 3. This has important consequences for
the asymmetry of the co-oscillating tides in the Dutch
tidal basins and estuaries, as will be shown in sections 4 and 5. The generation of higher harmonic tidal
constituents in coastal shelf seas and their influence
on sediment transport have also been discussed recently by PINGREE
et al. (1984).
So far friction has been neglected. However,frictional influences cannot be ignored in most coastal
seas and estuaries. The non-linearity of the friction
term causes, in the uniform systems considered
here, a larger frictional influence at Iow water than at
high water,yielding (slightly) larger flood than ebb velocities (see Fig. 10b).Another aspect is the retardation of the tidal velocity with respect to the water level
elevation. The maximum flood velocity occurs during
rising tide, the maximum ebb velocity during falling tide; the magnitude depends on the actual inclination
of the water surface. The formerly discussed steeper
rise of the tide and slower decrease caused by the
non-linear terms u6u/6x and 6u~/6xproduce, in combination with friction, a relatively larger flood velocity
and a relatively smaller ebb velocity. These effects
counteract and even dominate the Stokes drift, as
shown in Fig. 10c. The occurrence of larger maxi-
mum flood velocities than ebb velocities is a commonly observed feature in tidal rivers (at least at Iow
river runoff), see Fig. 2. It tends to favour landward
transport of sediment over seaward transport.
As mentioned in section 1, river runoff produces a
cross-sectional distribution of current velocities
which enhances seaward transport in the near surface part, but counteracts seaward transport in the
deeper parts of the channel. This phenomenon will
not be discussed here, but should be considered for
practical applications. Finally, it is noted that river runoff not only influences the cross-sectional distribution of currents. A distortion of the tidal wave may also
result, at least if river and tidal discharges are comparable (GODIN,1985). The main cause is the quadratic dependance of bottom shear stress on current
speed (HEATH,1981).However,the direction of the residual sediment transport is not strongly influenced:
at high river runoff seaward transport dominates
(MIGNIOT,1968).
4. DISTORTIONOF A STANDING TIDAL WAVE
In basins with a length I much shorter than the tidal
wavelength L or a constriction at a distance I from the
(m)
2
I
0
-,
-2
-
:V"
:lP
,
-,
\..,J
~.....
:~
-1
0
A
'p V
~
~ r-
1\
-,
Fig. 3. Distortion of the tidal wave propagating
'\..,
along the Dutch coast (RIJKSWATERSTAAT).
IT
ID
Dl
1L
\j
TIDAL ASYMMETRY AND ESTUARINE MORPHOLOGY
sea boundary, 1« L, the major part of the tidal wave
ISreflected at the head. The simplest schematization
IS a rectangular basin of constant depth. In the appendix the influence of the non-linear terms in the
continuity equation and the momentum equation is
demonstrated for such a simple geometry.
Without the friction term the tidal motion is almost
a standing wave, slightly distorted by the non-linear
terms d~u/dxand udu/dx. This distortion consists essentially of a relatively longer slack water period before ebb than before flood (Fig. 10d).The reason is that
changes in water level propagate faster at high water
than at Iowwater; a nearly horizontal water surface is
more readily reached and maintained. As a result the
water level inclination and corresponding flow acceleration is larger in magnitude at Iow water than at
high water.
The influence of the friction term is twofold. Firstly,
part of the energy of the incoming tidal wave is absorbed; thus the reflected wave is smaller than the incoming wave: the tidal motion has the character of a
partly progressive wave (lpPEN& HARLEMAN,
1966).
Secondly, the frictional influence is less around high
water than around Iowwater.The effect is opposite to
that of the other non-linear terms: around high water
(just before and after slack water) the magnitude of
the current is larger than around Iow water (see Fig.
We)
I du/dt I SBE> I du/dt I SBF
In a tidal basin with landward decreasing depth the
influence of friction is not the same when the tidal variation of the current velocity in a moving frame is
considered. If the decrease in channel depth on the
flood excursion of a fluid parcel exceeds the tidal range, then the friction experienced in the period around
high water slack (SBE) is larger than the friction experienced around Iow water slack (SBF).
Consequently,
123
faster around high water than around Iow water. The
time delay between high water at different locations
in the estuary is shorter than the delay between Iow
water. Similarly the time delay between SBE at different locations is smaller than the time delay between
SBF (dt +/dx < dt -/dx). In other words, the flood duration in the inner part of the estuary is shorter than
the ebb duration. Consequently, the maximum flood
velocity exceeds the maximum ebb velocity. This effect is enhanced when frictional energy dissipation is
important. In that case, water level variations are diffused through the estuary, rather than propagated.
As shown by LE BLOND(1978),diffusion speed depends even more strongly on water depth than wave
speed.
In the case of short tidal basins, distortion by nonlinear terms is rather small. The water motion has essentially a co-oscillating character with nearly
constant phase and water-levelthroughout the basin.
In fact the most important causes for tidal asymmetry
are (see also the studies of PETHICK,1980; BOON&
BVRNE, 1981):
- an asymmetry in the tidal boundary condition,
- a variation in the basin geometry with water-level.
In a co-oscillating tidal basin the current velocity
can be obtained by integration of the continuity equation (1):
E ~I
u==A s dt I x=o
(6)
Here ~ = JxlbdX is the storage surface of the basin.
This quantity, as well as the stream cross section As,
depends on the water level ~.
The influence of the tidal boundary condition ~(t)
on the current velocity follows directly from equation
(6). As mentioned already, a distortion of the tidal wave propagating over the shallow coastal shelf influences the tidal motion in the adjacent estuaries. For
example, the relatively fast rise and slow decrease of
the tidal wave propagating along the Dutch coast is
I du/dt I SBF> I du/dt I SBE
the major cause of the occurrence of a greater maxilor a fluid parcel moving with the tide: in a rectangu- mum current during flood than during ebb in the
lar basin with landward decreasing depth, a land- Dutch tidal basins (see Fig. 4). The boundary tide
ward residual transport of fine sediment is favoured. also affects the slack water periods before ebb and
The distortion of a tidal wave in basins of limited flood, but to a lesser degree. Here the major cause
length due to both non-linear propagation terms and of asymmetry is the tidal variation of the basin
friction has been studied by several authors: KREISS geometry.
(1957), HEATH(1981) and UNCLES(1981). They find
The time derivative at slack water of equation (6)
that the maximum flood velocity exceeds the maxi- yields:
mum ebb velocity, at least in the inner part of the
estuary (in the absence of river discharge). The readu I
==£
/)2~ I
(7)
dt I slack As dt2 I slack
son is again that water level changes propagate
J. DRONKERS
124
flot
lEvEl
CURRENT
CURRENT
DISCHARGE
VELOCITY
VELOCITY
[m'/s]
[m/si
[m]
.~
'10
.,
.05
0
0
FLOOD
[m/s]
".0
FLOOD
'0.5
0
.,
-0.5
. 2
-10
EBB
-0.5
EBB
.1.5
'15
.1.0
.2
.10
.1
.0.5
0.5
0
0
0
-0.5
.1
-0.5
-2
-1.0
.
2
50
.1
25
-1.0
[m'/sJ
210'
DISCHARGE
7510'
[m'fs]
1.10'
0
-110'
0
0
-1
-25
- 2
-50
75
.2
50
.1
25
0
0
[m/sI
.1.0
FLOOD
.0.5
0
EBB
- 1
-25
-2
.50
-0.5
0
-75
0
4
8
12
4
8
12
11\
20 thOU'>]
Fig. 5. Field observation of geometry-induced tidal wavedeformation in type 1 tidal basins.
16 {hours]
stream cross-section. This characteristic favours a
Fig. 4. Tidal variation of discharge/current velocity (-)
and
longer slack water period before ebb, I du/dt I SBE
water level (- - -) in Dutch tidal basins (RIJKSWATERSTAAT,
unpublished data).
< I du/dt I SBF'Examples are shown in Fig. 6.
On the basis of this expression one may distinguish
between two types of tidal basins:
(1)The relative increase of stream cross-section As
with increasing water-level is smaller than the relative
increase of storage surface. The corresponding tidal
basins have deep channels (h > > a) and/or high tidal
flat areas (on the average above mean sea level).
This characteristic favours a longer slack water period before flood than before ebb, I du/dt I SBF <
I du/dt I SBE'Examples are shown in Fig. 5.
(2) The relative increase of stream cross-section
with increasing water-levelis greater than the relative
increase of storage surface. The corresponding tidal
basins have shallow channels (depth at most a few times larger than the tidal amplitude) and/or Iow tidal
flat areas, which at high water become part of the
The above implies that the tidal variation of the wetted geometry affects the residual transport of fine sediment. In the type 1 tidal basin, residual export of fine sediment is favoured. In the type 2 tidal basins, residual import of fine marine sediment can be expected and retention of the fine fluvial sediment discharged at the head.
It has been noted by SPEER& AUBREY(1985)and
by BOON& BYRNE(1981)that large tidal flats may enhance the maximum ebb current. The tidal wave propagates faster in the channels than on the tidal flats.
Therefore, the decrease in water-level during ebb takes place later on the tidal flat than in the channel.
This leads to an important water-level inclination (see
Fig. 1b) and a strong current during the last stage of
the ebb period. A net seaward flux of coarse suspended sediment may result.
TIDAL ASYMMETRY AND ESTUARINE MORPHOLOGY
CURRENT
VELOCITY
5. TIDAL ASYMMETRY AND SEDIMENT
TRANSPORT IN THE WADDEN SEA AND
THE EASTERN SCHELDT
[mls]
.
125
1.0
.05
0
- 0 5
.05
0
- 0.5
.1 0
.05
0
- 0.5
- 1.0
l
0
4
8
12
[hours]
Fig. 6. Field observations of geometry-induced tidal wave
deformation in type 2 tidal basins.
In this section field observations in two tidal basins in
the Netherlands will be discussed as an example of
the more general principles established previously. In
both tidal basins fresh water inflow is extremely
small. The current distribution is mostly tide-induced;
wind forcing is a secondary effect.
The morphology of the Wadden Sea Ameland area
and the Eastern Scheldt basin is shown in Fig. 7, on
the same scale. Both tidal basins are much shorter
that the tidal wave length. In the Eastern Scheldt large deep channels bounded by levees lead to tidal
flats situated in the landward part; in the Wadden Sea
a sequence of large, medium and small channels rapidly lead from deep to shallow regions. According to
these morphological differences the tidal wave is distorted differently in both systems. This is most pronounced for the slack water periods. In Fig. 8 the bathymetry of the two tidal basins is represented by the
curves showing the surface at different depths. In the
Wadden Sea the relative increase of the volume with
OOSTERSCHELDE
.
'
~
::.'~-'
~
::>i
\
""-tf-,:.<4"-,~~
j/?
0
~
5
::11
10 km
I
>~,'
WADDEN SEA AMELAND AREA
'~I.~i~ft~~>:,r~~*~l':d
10m DEPTH LINE
. MEAN lOW
WATER LINE
Fig. 7. Morphology of the Wadden Sea Ameland area and the Eastern Scheldt basin.
126
J. DRONKERS
(0)
U( t)
,
[m"J
-"W
t [hOU," )
--LW
(m]
-,
( b)
WADDEN SEA AMELAND
AREA LOCATION 8
"W
[m]
I
I
"W~~---
"- "-
_10,1
U (t)
"-"-
'\
/
LW---
\
[m/')
\
/
D
.--/
-,
'.10'
L
As
10'
EASTERN
['O'm']
SC"ELDT
LOCATION
WS
Fig. 9. Current velocity variation in the Wadden Sea Ameland area (location 8) and in the Eastern Scheldt basin (location WS).
L . jb(')dX
-
-
-
-
WADDENSEA.AMELAND AREA
OOSTERSC"ELDE
butes to the inequality Idu/dt I SBE< I du/dt I SBF'In
the Eastern Scheldt the boundarytide contributes to
the opposite inequality.
(0)
SURFACE
AS A FUNCTION
OF DEPT"
(b)
RATIO OF STORAGE
AREA ~ AND STREAM
CROSS- SECTION
A.
Large tidal flats are presentin the Wadden Sea and
in
the Eastern Scheldt. The major part is covered onFig. 8. Surface as a function of depth in the Wadden Sea
Ameland area and the Eastern Scheldt basin; the ratio "LIAs ly around high water. In this period bottom material
as a function of tidal level.
(fine and coarse sediment)can be brought in suspension and subsequently be transported by ebb currents. Thus the residual landwardtransport by tidal
tidal level is greater than the relative increase of sur- currents of both coarse and finesediment is counterface.The ratio '£./Asat high wateris smallerthan at acted. In the Wadden Seathe fine sediment fraction
Iow water. According to equation (7) one may expect, in the intertidal areas is much larger than in the
in theWaddenSea Idu/dt I SBE< I du/dt I SBF'In the Eastern Scheldt. Therefore,one may expect that the
Eastern Scheldt the opposite inequality should hold. influence of wind wavesaffectsmorestrongly the fine
Fig. 9 shows the agreement of these predictions with sediment transport in the WaddenSea.
field data.
One may also expect that a residual landward
The peak flood discharge slightly exceeds the transport of sediment prevailsunder calm weather
peak ebb discharge in both the Wadden Sea and the conditions, while seawardtransportdominates under
Eastern Scheldt for an average tide, see Fig. 4. The storm conditions.
extent of tidal flat areas is insufficient to cancel the
Field observations showthe following:In the Wadflood dominance due to the faster rise and slowerde- den Sea, bottom soundings overa period of 20 years
crease of the tide along the Dutch coast (see Fig. 3). indicate an average bottomaccretionof 0.5-1cm.a-1
(Locally ebb currents may dominate flood currents (DE BOER& VISSER,1981).In the Eastern Scheldt,
due to the presence of ebb and flood dominated over the period 1872-1974an averageerosion of 1
channels). Therefore, flood transport of coarse sedi- cm.a -1 is found, VANDENBERG
(1984).However, the
ment should in principle exceed ebb transport.
net erosion rate has decreasedin the last decade
The tidal boundary curves (Fig. 3) further show and does not exceed a few mm.a-1. An extensive
that the second derivative d2~/dt2presents a different sediment transport measurementcampaigncovering
asymmetry in the Wadden Sea and in the Eastern a whole year, but excluding stormperiods yielded a
Scheldt. In the Wadden Sea
Id2~/dt21
HW net export of sediment (essentiallyfine sediment)
< I d2Vdt2I LW'This means that the distortion of the corresponding to an averageerosion of 0 to 3
tidal wave running along the Dutch coast also contri- cm.a -1 (DRONKERS,1985).
BAT"YMETRY
""
OOSTERSC"ELDE
AND AMELAND
TIDAL
SYSTEMS,
TIDAL ASYMMETRY AND ESTUARINE MORPHOLOGY
TIDE
LEVEL
CURRENT
VELOCITY
rections of the residual fluxes are in agreement.
- the magnitude of the residual flux of fine sediment is strongly influenced by wind waves, and
storms, especially in the Wadden Sea (see also WIN-
[m/s]
[m]
,.0
1.D
.05
.0.0
KELMOLEN& VEENSTRA,1980; DRONKERS,1985)
.0..0
-.0.5
-,.0
-'.0
/-
/'
"-
::~
.0..0
"-
-.0.5
~".0..0
~2
.05
1..0
.05
DD
DD
-.0.5
-0.5
-10
.05
DD
.0..0
-.05
-,.0
-.05
1.0
.05
.0.5
DD
.0..0
-.0.5
-.05
-1..0
Fig. 10. Time
velocity (-)
t = a
cos.dt
variation
of water-level
in a uniform
at
x = O.
The
channel
(-
- -)
with open
amplitude
and
sea
a = 1 m,
127
current
boundary.
the
depth
h = 10 m; for damped waces the friction coefficient F =
0.002 m.s-1.a. Progressive non-linear wave without friction
at x= 100km.b. Progressivedampedwave,withouthnon-
linear terms du~/du,udu/dx.c. Progressivedampednonlinear wave. d. Standing non-linear wave without friction at
x = 25 km in a canal of length I = 50 km. e. Standing damped
wave, without non-linear terms duVdx, udu/dx.
The expression (1) for tide-induced residual transport of fine sediment yields a net import corresponding to an average bottom accretion of 1-6 cm.a-1
for the Wadden Sea, and for the Eastern Scheldt a
net export corresponding to an average bottom erosion of 0 to 0.5 mm.a -1.
A comparison of the field observations with the
predictions for fine sediment transport, taking into
account the qualitative tendencies that have been indicated for storm affects and for the residual transport of coarse sediment, allows the conclusion that:
- for fine sediment, the observed and predicted di-
- for coarse suspended sediment the landward direction of the tide-induced residual flux in the Wadden Sea agrees with observations. In the Eastern
Scheldt an important contribution to the residual
transport of coarse suspended sediment seems to be
provided by the action of wind waves in combination
with tidal transport. The stronger impact of wind waves in the Eastern Scheldt compared to the Wadden
Sea can also be traced to the different orientations of
these basins with respect to the average wind direction: in the Eastern Scheldt the direction of the main
channel axis fits the average wind direction more
closely.
The field observations indicate that a morphological equilibrium in the Wadden Sea and the Eastern
Scheldt is not yet reached in the period under consideration. However, different processes involving
wind- and tide-induced processes yield mutually
counteracting residual fluxes of the same order of
magnitude. This implies that small morphological
changes can alter the total residual transport: the actual state cannot be very far from morphological equilibrium. In fact, the major external change in the
Wadden Sea Ameland area during the past decades
is an increase of the mean sea level of 0.2 cm.a -1 on
the average. If the sea level stabilizes, the most probable evolution of the Wadden Sea is a further accretion of the tidal flats until a quasi-equilibrium between
tide- and wind-induced sediment fluxes is reached.
In the Rhine-Meuse-Scheldt delta successive engineering works have caused a considerable increase
of the tidal prism in the Eastern Scheldt during the
period 1872-1974and in particular between 1965 and
1970. The tidal energy dissipation in the Eastern
Scheldt increased. Erosion dominated over sedimentation and ebb sediment fluxes over flood sediment
fluxes, especially in periods shortly after maninduced alterations of the hydraulic regime. In the future a storm surge barrier in the mouth of the Eastern
Scheldt, presently under construction, will decrease
again the tidal influence and the sediment motion in
this basin. A slow infill with sediment is expected.
6. SUMMARY AND CONCLUSIONS
1)The most pertinent features of tidal wave distortion
for residual sediment transport are:
- a difference between the slack water periods
before ebb and flood ( I du/dt I SBE* I du/dt I SBF)'
128
J. ORONKERS
whiCh affects in particular the residual transport of
the fine fraction of the suspended load.
- a difference between the maximum currents during ebb and flood (u~bt"* uH6~~),
which especially affects the residual transport of the coarse fraction of
the suspended load.
2) In regularly shaped basins (no important width
variation with water-level) and in the absence of river
inflow, the tidal wave tends to be distorted such that
u~;;d > u~b~x,I du/dt I SBE< I du/dt I SBF'This distortion is manifest in the inner part of both long and short
tidal basins (compared to the tidal wave length). It will
cause a sediment infill of estuaries in periods of small
river discharge.
3) In irregularly shaped estuaries (meandering and
braiding channel system, tidal flats) the tidal current
variation is influenced by the geometry. Twotypes of
geometry can be distinguished:
- shallow channels/landward decreasing depth, tidal flats below mean sea level.
- deep channels throughout, tidal flats above mean
sea level.
In the first case the slack water period before ebb
will exceed the slack water period before flood: a residual import of fine sediment is favoured. In the second case the inverse situation occurs.
In addition, large tidal flats cause an enhancement
of the maximum ebb current and may, therefore, induce a seaward flux of coarse suspended sediment.
4) The distortion of the tidal wave propagating on
the coastal shelf also induces an ebb-flood asymmetry in the tidal current variation inside adjacent bays
and estuaries. If the rise of the coastal water-level is
faster than the decrease, in a co-oscillating tidal basin uH6~~>u~bt"and a sediment infill is favoured.
5) Wind induced resuspension of sediment may
also present a tidal asymmetry. In many estuaries the
major part of the fine grained tidal flats are covered
around high water only. In such cases wind induced
resuspension leads to an export of fine sediment.
AVOINE,J., G.P.ALLEN,M. NICHOLS,
J.C. SALOMON
& C. LARSONNEUR,
1981.Suspended-sediment transport in the
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119-137.
BERGVANDEN,J.R., 1984. Morphological changes of the
ebb-tidal delta of the Oosterschelde during recent decades, Geologie Mijnb. 63: 363-375
BIGGs,R.B., 1978.Coastal bays. In: RA DAVls.Coastal sedimentary environments. Springer Verlag: 1-420.
BOER,M. DE& G.C. VISSER,1981.Erosie en sedimentatie in
the westelijke Waddenzee. Nota WWKZ- 80.HO01,
Rijkswaterstaat, The Netherlands: 1-38.
BooN, J.D. & R.J. BVRNE,1981.On basin hypsometry and
the morphodynamic response of coastal inlet
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BRuuN, P., 1978. Stability of tidal inlets. Elsevier Amsterdam: 1-510.
CHASE,R.R.P., 1979. Settling behavior of natural aquat,ic
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COMOV,M., 1881.Etude practique sur les marE~esfluviales
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CREUTZBERG,
F. & H. POSTMA,
1979.An experimental approach to the distribution of mud in the southern North
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ticulate and dissolved organic and inorganic matter
between a salt marsh and the Ems-Dollard estuary,
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DRoNKERs,
J., 1985.Tide-induced residual transport of fine
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OVER,K.R., 1980. Velocity profiles over a rippled bed and
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EINSTEIN,
H.A & R.B. KRONE,1962. Experiments to determine modes of cohesive transport in salt water, J. geophys. Res. 67: 1451-1461.
EISMA,D., J. KALF& M. VEENHUIS,
1980. The formation of
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FESTA,
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1978.Turbidity maxima in partially
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POSTMA,
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130
J. DRONKERS
APPENDIX
The one-dimensional tidal equations (1-2) are solved for uniform systems (constant depth, width) to first order in a/h, according to standard methods as indicated, e.g. in LAMB(1932). For distinction between the impacts of the different non-linear
terms, multiplication factors f1, f2 and f3 have been introduced. The equations read
cg+h~ +f d~u=0
dt
dx 1 dx
du
-
dt
du
d~
+f u- +g-+F2 dx
dx
(A1)
u
h
. u~
-f F- =0
(A2)
3 h
Solutions are presented for systems with a prescribed water-levelvariation at the ocean boundary: W)=a cos at. The solutions presented in HEATH
(1980)for the same tidal equations (but with a quadratic friction law) satisfy different boundary conditions. The qualitative features of the solutions appear to be similar, however.
SOLUTIONS FOR PROGRESSIVE WAVES
1. Without friction (F=O), see Fig. 10a.
~= a[ cos(at-kx)-(f1 +2f2)i ~kX sin(2at-2 kX)]
u= ~
kh
)]
(
[ cOS(at-kX)-
~ 1 + 1(2f2 -f,)cos(2at-2kx)+
h 2 8
Here the wave number k is given by k = a/~.
1(2f2
+f1)kx sin(2at-2 kX)
4
The validity of these solutions is restricted to
X<'<:
h/ka
(LAMB,
2. Non-linear friction (F> 0, f3= 1) but no other non-linear terms (f, = f2= 0), see Fig. 10b.
~=
~
u=
~
[
k2R (1 - e2klX)+ 1a ei(at- kx) - ~ e2i(at - kx) - e2i(at - k'X) + c.c.
2h I k I 2
2
2h
.
2h
[lei(at-kX)
k
- ~
2h
(
(le2i(at-kX)_1
k
k'
)
e2i(at-k'X) +c.c.
)
]
]
Here k and k' are complex wave numbers, kR=Re k>O, kl=lm k<O,
ala-if)
k2= ~
gh
3. Non-linear
2
k'=~
ala-if
'
)
gh
friction (F > 0, f3 = 1) and non-linear terms (f 1 = f2 = 1), see Fig. 1Oc.
~=B(1-e2kiX)+ ~a[ ei(at-kx) +A( e2i(at-kx) -e2i(at-k'X)) +c.c.]
u=Ce2klX + aa
2h
+ (A[ lei(at-kX)
(
k
Here A= - ~ (2 f + 3ia),
2F h
~)
1e2i(at-kX)- ~ e2i(at-k'X) +c.c.
4h2 k
a2a2
B=
4gh2 I k 12 (1-2fW
)
k'
F kR
1<;)'
C= -
]
aa2kR
2 h2 I k I 2
1932)
TIDAL ASYMMETRY AND ESTUARINE MORPHOLOGY
SOLUTIONS
FOR STANDING WAVES
A basin is considered with a closed end at x=1 and an ocean boundary at x=o.
4. Non-linear
terms (11' 12",0), no Iriction (F = 0), see Fig. 10d.
a
~=
-cos kl
u-- ~
[cos k(l-x)
kh cos kl
1 a2
]
cos at-(I, +212)-8 h2 kx cos 2k(l-x)cos 2 at
[sin k(l-x)sin at-~h (~(2f2-ll)+
16
[
I+(I-X)Cot 2k(I-X) sin 2k(l-x)sin 2at
~(212+1,)k
D
]
8
5. Non-linear Iriction (F>O, 13= 1), no other non-linear terms (I, =12=0), see Fig. 10e.
Faa2
~=
.
k2, cos 2kRI-cos 2kR(I-x) +k2R cosh 2k,l-cosh 2k,(I-x)
4g h3 I k 12
[
+~a COSk(l-x) eiat2
u=i~
2h
cos kl
kRk, cos 2kRI+cosh 2k,1
(
a
COS2k(l-x)4h COS2kl
Sin 2k(l-x) k(l-x) eiata
[Sin
k cos kl
4h cos2 kl (
k
]
cos 2kl cos 2k'(I-X» e2iat+c.c.
)
cos 2k'l
cos 2kl sin 2k'(I-X) e2iat+c.c.
cos 2k'l
k'
)
]
131